Question

Let \( f_n: [0, 10] \to \mathbb{R} \) be given by \( f_n(x) = n x^3 e^{-n x} \) for \( n = 1, 2, 3, \dots \). Consider the following statements: P: \( (f_n) \) is equicontinuous on \( [0, 10] \). 
Q: \( \sum_{n=1}^{\infty} f_n \) does NOT converge uniformly on \( [0, 10] \). Then:

• A. both P and Q are TRUE
• B. P is TRUE and Q is FALSE
• C. P is FALSE and Q is TRUE
• D. both P and Q are FALSE

Hint:

For a series to converge uniformly, the terms must satisfy the Weierstrass M-test or similar conditions, which is not the case here.

Answer 1

The Correct Option is B

Step 1: Analyze statement P.
For a sequence of functions \( (f_n) \) to be equicontinuous, the change in \( f_n(x) \) must be uniformly small for all \( x \) when \( x \) is close to some fixed value. - We compute \( f_n'(x) \), and check if the sequence satisfies the definition of equicontinuity. - After verification, it is found that \( (f_n) \) is equicontinuous on \( [0, 10] \), hence statement P is TRUE.

Step 2: Analyze statement Q.
For uniform convergence of a series \( \sum_{n=1}^{\infty} f_n(x) \), we need to check if the partial sums converge uniformly on the interval. - In this case, the series does not converge uniformly on \( [0, 10] \), so statement Q is FALSE.

Final Answer: (B) P is TRUE and Q is FALSE